Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Percentage Accurate: 51.2% → 99.9%

Time: 11.0s

Alternatives: 11

Speedup: 3.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 51.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, y\_m \cdot 2\right)\\ \frac{y\_m \cdot 2 + x}{t\_0 \cdot \frac{t\_0}{x + y\_m \cdot -2}} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (hypot x (* y_m 2.0))))
   (/ (+ (* y_m 2.0) x) (* t_0 (/ t_0 (+ x (* y_m -2.0)))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = hypot(x, (y_m * 2.0));
	return ((y_m * 2.0) + x) / (t_0 * (t_0 / (x + (y_m * -2.0))));
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = Math.hypot(x, (y_m * 2.0));
	return ((y_m * 2.0) + x) / (t_0 * (t_0 / (x + (y_m * -2.0))));
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = math.hypot(x, (y_m * 2.0))
	return ((y_m * 2.0) + x) / (t_0 * (t_0 / (x + (y_m * -2.0))))

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = hypot(x, Float64(y_m * 2.0))
	return Float64(Float64(Float64(y_m * 2.0) + x) / Float64(t_0 * Float64(t_0 / Float64(x + Float64(y_m * -2.0)))))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	t_0 = hypot(x, (y_m * 2.0));
	tmp = ((y_m * 2.0) + x) / (t_0 * (t_0 / (x + (y_m * -2.0))));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(y$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[(N[(y$95$m * 2.0), $MachinePrecision] + x), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 / N[(x + N[(y$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 55.1%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 55.1%

      \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. difference-of-squares 55.1%

      \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   3. *-commutative 55.1%

      \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. associate-*r* 55.1%

      \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   5. sqrt-prod 55.1%

      \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   6. sqrt-unprod 29.2%

      \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   7. add-sqr-sqrt 41.2%

      \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   8. metadata-eval 41.2%

      \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   9. *-commutative 41.2%

      \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   10. associate-*r* 41.2%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   11. sqrt-prod 41.2%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   12. sqrt-unprod 29.2%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   13. add-sqr-sqrt 55.1%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   14. metadata-eval 55.1%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

4. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
5. Step-by-step derivation

   1. *-commutative 55.1%

      \[\leadsto \frac{\color{blue}{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. associate-/l* 56.2%

      \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{x + y \cdot 2}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   3. +-commutative 56.2%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{y \cdot 2 + x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. fma-define 56.2%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   5. *-commutative 56.2%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{y \cdot \left(y \cdot 4\right)}} \]

   6. associate-*r* 56.2%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot y\right) \cdot 4}} \]

   7. metadata-eval 56.2%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \left(y \cdot y\right) \cdot \color{blue}{\left(2 \cdot 2\right)}} \]

   8. swap-sqr 56.2%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

   9. add-sqr-sqrt 56.2%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}}} \]

   10. hypot-undefine 56.2%

       \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

   11. hypot-undefine 56.2%

       \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   12. unpow2 56.2%

       \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

6. Applied egg-rr 56.2%

   \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]
7. Step-by-step derivation

   1. associate-*r/ 55.1%

      \[\leadsto \color{blue}{\frac{\left(x - y \cdot 2\right) \cdot \mathsf{fma}\left(y, 2, x\right)}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

   2. unpow2 55.1%

      \[\leadsto \frac{\left(x - y \cdot 2\right) \cdot \mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   3. frac-times 99.9%

      \[\leadsto \color{blue}{\frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   4. *-commutative 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   5. clear-num 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x - y \cdot 2}}} \]

   6. frac-times 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right) \cdot 1}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x - y \cdot 2}}} \]

   7. *-commutative 100.0%

      \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(y, 2, x\right)}}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x - y \cdot 2}} \]

   8. *-un-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x - y \cdot 2}} \]

   9. sub-neg 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{\color{blue}{x + \left(-y \cdot 2\right)}}} \]

   10. distribute-rgt-neg-in 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + \color{blue}{y \cdot \left(-2\right)}}} \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + y \cdot \color{blue}{-2}}} \]

8. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + y \cdot -2}}} \]
9. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + y \cdot -2}} \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + y \cdot -2}} \]
11. Add Preprocessing

Alternative 2: 66.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := x - y\_m \cdot 2\\ t_1 := \mathsf{hypot}\left(x, y\_m \cdot 2\right)\\ \mathbf{if}\;x \leq 1.25 \cdot 10^{-126}:\\ \;\;\;\;\frac{t\_0}{t\_1} \cdot \left(1 + 0.5 \cdot \frac{x}{y\_m}\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+134}:\\ \;\;\;\;t\_0 \cdot \frac{y\_m \cdot 2 + x}{{t\_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (- x (* y_m 2.0))) (t_1 (hypot x (* y_m 2.0))))
   (if (<= x 1.25e-126)
     (* (/ t_0 t_1) (+ 1.0 (* 0.5 (/ x y_m))))
     (if (<= x 6e+134)
       (* t_0 (/ (+ (* y_m 2.0) x) (pow t_1 2.0)))
       (+ 1.0 (* -8.0 (* (/ y_m x) (/ y_m x))))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = x - (y_m * 2.0);
	double t_1 = hypot(x, (y_m * 2.0));
	double tmp;
	if (x <= 1.25e-126) {
		tmp = (t_0 / t_1) * (1.0 + (0.5 * (x / y_m)));
	} else if (x <= 6e+134) {
		tmp = t_0 * (((y_m * 2.0) + x) / pow(t_1, 2.0));
	} else {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = x - (y_m * 2.0);
	double t_1 = Math.hypot(x, (y_m * 2.0));
	double tmp;
	if (x <= 1.25e-126) {
		tmp = (t_0 / t_1) * (1.0 + (0.5 * (x / y_m)));
	} else if (x <= 6e+134) {
		tmp = t_0 * (((y_m * 2.0) + x) / Math.pow(t_1, 2.0));
	} else {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = x - (y_m * 2.0)
	t_1 = math.hypot(x, (y_m * 2.0))
	tmp = 0
	if x <= 1.25e-126:
		tmp = (t_0 / t_1) * (1.0 + (0.5 * (x / y_m)))
	elif x <= 6e+134:
		tmp = t_0 * (((y_m * 2.0) + x) / math.pow(t_1, 2.0))
	else:
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(x - Float64(y_m * 2.0))
	t_1 = hypot(x, Float64(y_m * 2.0))
	tmp = 0.0
	if (x <= 1.25e-126)
		tmp = Float64(Float64(t_0 / t_1) * Float64(1.0 + Float64(0.5 * Float64(x / y_m))));
	elseif (x <= 6e+134)
		tmp = Float64(t_0 * Float64(Float64(Float64(y_m * 2.0) + x) / (t_1 ^ 2.0)));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = x - (y_m * 2.0);
	t_1 = hypot(x, (y_m * 2.0));
	tmp = 0.0;
	if (x <= 1.25e-126)
		tmp = (t_0 / t_1) * (1.0 + (0.5 * (x / y_m)));
	elseif (x <= 6e+134)
		tmp = t_0 * (((y_m * 2.0) + x) / (t_1 ^ 2.0));
	else
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(x - N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[x ^ 2 + N[(y$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[x, 1.25e-126], N[(N[(t$95$0 / t$95$1), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e+134], N[(t$95$0 * N[(N[(N[(y$95$m * 2.0), $MachinePrecision] + x), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.25000000000000001e-126

   1. Initial program 54.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 54.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 54.5%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 54.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 54.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 54.5%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 29.0%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 39.5%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 39.5%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 39.5%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 39.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 39.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 29.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 54.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 54.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 54.5%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 54.5%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 55.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 55.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-define 55.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 55.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-define 55.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. *-commutative 55.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. sqrt-prod 29.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y} \cdot \sqrt{y \cdot 4}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 29.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{y} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{4}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 29.7%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{y} \cdot \left(\sqrt{y} \cdot \color{blue}{2}\right)\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*l* 29.7%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right) \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 55.9%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y} \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around inf 33.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   if 1.25000000000000001e-126 < x < 5.99999999999999993e134

   1. Initial program 85.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 85.9%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 85.9%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 85.9%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 85.9%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 85.9%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 43.7%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 66.9%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 66.9%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 66.9%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 66.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 66.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 43.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 85.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 85.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 85.9%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 85.9%

         \[\leadsto \frac{\color{blue}{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. associate-/l* 86.1%

         \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{x + y \cdot 2}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      3. +-commutative 86.1%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{y \cdot 2 + x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. fma-define 86.1%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. *-commutative 86.1%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{y \cdot \left(y \cdot 4\right)}} \]

      6. associate-*r* 86.1%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot y\right) \cdot 4}} \]

      7. metadata-eval 86.1%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \left(y \cdot y\right) \cdot \color{blue}{\left(2 \cdot 2\right)}} \]

      8. swap-sqr 86.1%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      9. add-sqr-sqrt 86.1%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}}} \]

      10. hypot-undefine 86.1%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      11. hypot-undefine 86.1%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

      12. unpow2 86.1%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

   6. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]
   7. Step-by-step derivation

      1. fma-undefine 99.9%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + y \cdot -2}} \]

   8. Applied egg-rr 86.1%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{y \cdot 2 + x}}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}} \]

   if 5.99999999999999993e134 < x

   1. Initial program 5.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 5.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 5.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 82.4%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 82.4%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. pow2 82.4%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 88.8%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 88.8%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-126}:\\ \;\;\;\;\frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+134}:\\ \;\;\;\;\left(x - y \cdot 2\right) \cdot \frac{y \cdot 2 + x}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, y\_m \cdot 2\right)\\ \frac{y\_m \cdot 2 + x}{t\_0} \cdot \frac{x - y\_m \cdot 2}{t\_0} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (hypot x (* y_m 2.0))))
   (* (/ (+ (* y_m 2.0) x) t_0) (/ (- x (* y_m 2.0)) t_0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = hypot(x, (y_m * 2.0));
	return (((y_m * 2.0) + x) / t_0) * ((x - (y_m * 2.0)) / t_0);
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = Math.hypot(x, (y_m * 2.0));
	return (((y_m * 2.0) + x) / t_0) * ((x - (y_m * 2.0)) / t_0);
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = math.hypot(x, (y_m * 2.0))
	return (((y_m * 2.0) + x) / t_0) * ((x - (y_m * 2.0)) / t_0)

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = hypot(x, Float64(y_m * 2.0))
	return Float64(Float64(Float64(Float64(y_m * 2.0) + x) / t_0) * Float64(Float64(x - Float64(y_m * 2.0)) / t_0))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	t_0 = hypot(x, (y_m * 2.0));
	tmp = (((y_m * 2.0) + x) / t_0) * ((x - (y_m * 2.0)) / t_0);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(y$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[(N[(N[(y$95$m * 2.0), $MachinePrecision] + x), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(x - N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 55.1%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 55.1%

      \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. difference-of-squares 55.1%

      \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   3. *-commutative 55.1%

      \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. associate-*r* 55.1%

      \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   5. sqrt-prod 55.1%

      \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   6. sqrt-unprod 29.2%

      \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   7. add-sqr-sqrt 41.2%

      \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   8. metadata-eval 41.2%

      \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   9. *-commutative 41.2%

      \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   10. associate-*r* 41.2%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   11. sqrt-prod 41.2%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   12. sqrt-unprod 29.2%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   13. add-sqr-sqrt 55.1%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   14. metadata-eval 55.1%

       \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

4. Applied egg-rr 55.1%

   \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 55.0%

      \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

   2. times-frac 56.4%

      \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

   3. +-commutative 56.4%

      \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   4. fma-define 56.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   5. add-sqr-sqrt 56.4%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. hypot-define 56.4%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   7. *-commutative 56.4%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   8. sqrt-prod 30.0%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y} \cdot \sqrt{y \cdot 4}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   9. sqrt-prod 30.0%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{y} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{4}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   10. metadata-eval 30.0%

       \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{y} \cdot \left(\sqrt{y} \cdot \color{blue}{2}\right)\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   11. associate-*l* 30.0%

       \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right) \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   12. add-sqr-sqrt 56.4%

       \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y} \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]
7. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + y \cdot -2}} \]

8. Applied egg-rr 99.9%

   \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]
9. Add Preprocessing

Alternative 4: 66.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := x - y\_m \cdot 2\\ \mathbf{if}\;x \leq 1.25 \cdot 10^{-126}:\\ \;\;\;\;\frac{t\_0}{\mathsf{hypot}\left(x, y\_m \cdot 2\right)} \cdot \left(1 + 0.5 \cdot \frac{x}{y\_m}\right)\\ \mathbf{elif}\;x \leq 10^{+136}:\\ \;\;\;\;\frac{\left(y\_m \cdot 2 + x\right) \cdot t\_0}{x \cdot x + y\_m \cdot \left(y\_m \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (- x (* y_m 2.0))))
   (if (<= x 1.25e-126)
     (* (/ t_0 (hypot x (* y_m 2.0))) (+ 1.0 (* 0.5 (/ x y_m))))
     (if (<= x 1e+136)
       (/ (* (+ (* y_m 2.0) x) t_0) (+ (* x x) (* y_m (* y_m 4.0))))
       (+ 1.0 (* -8.0 (* (/ y_m x) (/ y_m x))))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = x - (y_m * 2.0);
	double tmp;
	if (x <= 1.25e-126) {
		tmp = (t_0 / hypot(x, (y_m * 2.0))) * (1.0 + (0.5 * (x / y_m)));
	} else if (x <= 1e+136) {
		tmp = (((y_m * 2.0) + x) * t_0) / ((x * x) + (y_m * (y_m * 4.0)));
	} else {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = x - (y_m * 2.0);
	double tmp;
	if (x <= 1.25e-126) {
		tmp = (t_0 / Math.hypot(x, (y_m * 2.0))) * (1.0 + (0.5 * (x / y_m)));
	} else if (x <= 1e+136) {
		tmp = (((y_m * 2.0) + x) * t_0) / ((x * x) + (y_m * (y_m * 4.0)));
	} else {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = x - (y_m * 2.0)
	tmp = 0
	if x <= 1.25e-126:
		tmp = (t_0 / math.hypot(x, (y_m * 2.0))) * (1.0 + (0.5 * (x / y_m)))
	elif x <= 1e+136:
		tmp = (((y_m * 2.0) + x) * t_0) / ((x * x) + (y_m * (y_m * 4.0)))
	else:
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(x - Float64(y_m * 2.0))
	tmp = 0.0
	if (x <= 1.25e-126)
		tmp = Float64(Float64(t_0 / hypot(x, Float64(y_m * 2.0))) * Float64(1.0 + Float64(0.5 * Float64(x / y_m))));
	elseif (x <= 1e+136)
		tmp = Float64(Float64(Float64(Float64(y_m * 2.0) + x) * t_0) / Float64(Float64(x * x) + Float64(y_m * Float64(y_m * 4.0))));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = x - (y_m * 2.0);
	tmp = 0.0;
	if (x <= 1.25e-126)
		tmp = (t_0 / hypot(x, (y_m * 2.0))) * (1.0 + (0.5 * (x / y_m)));
	elseif (x <= 1e+136)
		tmp = (((y_m * 2.0) + x) * t_0) / ((x * x) + (y_m * (y_m * 4.0)));
	else
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(x - N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.25e-126], N[(N[(t$95$0 / N[Sqrt[x ^ 2 + N[(y$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+136], N[(N[(N[(N[(y$95$m * 2.0), $MachinePrecision] + x), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * N[(y$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.25000000000000001e-126

   1. Initial program 54.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 54.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 54.5%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 54.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 54.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 54.5%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 29.0%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 39.5%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 39.5%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 39.5%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 39.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 39.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 29.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 54.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 54.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 54.5%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 54.5%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y} \cdot \sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      2. times-frac 55.8%

         \[\leadsto \color{blue}{\frac{x + y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}}} \]

      3. +-commutative 55.8%

         \[\leadsto \frac{\color{blue}{y \cdot 2 + x}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      4. fma-define 55.8%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      5. add-sqr-sqrt 55.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      6. hypot-define 55.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot 4\right) \cdot y}\right)}} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      7. *-commutative 55.8%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      8. sqrt-prod 29.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{y} \cdot \sqrt{y \cdot 4}}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      9. sqrt-prod 29.7%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{y} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{4}\right)}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      10. metadata-eval 29.7%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \sqrt{y} \cdot \left(\sqrt{y} \cdot \color{blue}{2}\right)\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      11. associate-*l* 29.7%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right) \cdot 2}\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      12. add-sqr-sqrt 55.9%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \color{blue}{y} \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\sqrt{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

   7. Taylor expanded in y around inf 33.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{x}{y}\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \]

   if 1.25000000000000001e-126 < x < 1.00000000000000006e136

   1. Initial program 85.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 85.9%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 85.9%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 85.9%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 85.9%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 85.9%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 43.7%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 66.9%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 66.9%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 66.9%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 66.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 66.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 43.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 85.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 85.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 85.9%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 1.00000000000000006e136 < x

   1. Initial program 5.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 5.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 5.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 82.4%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 82.4%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. pow2 82.4%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 88.8%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 88.8%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-126}:\\ \;\;\;\;\frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \left(1 + 0.5 \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 10^{+136}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, 2, x\right)}{y\_m \cdot -2 + x \cdot \left(-1 - \frac{x}{y\_m}\right)}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(y\_m \cdot 2 + x\right) \cdot \left(x - y\_m \cdot 2\right)}{x \cdot x + y\_m \cdot \left(y\_m \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= x 1.3e-126)
   (/ (fma y_m 2.0 x) (+ (* y_m -2.0) (* x (- -1.0 (/ x y_m)))))
   (if (<= x 1.5e+136)
     (/
      (* (+ (* y_m 2.0) x) (- x (* y_m 2.0)))
      (+ (* x x) (* y_m (* y_m 4.0))))
     (+ 1.0 (* -8.0 (* (/ y_m x) (/ y_m x)))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (x <= 1.3e-126) {
		tmp = fma(y_m, 2.0, x) / ((y_m * -2.0) + (x * (-1.0 - (x / y_m))));
	} else if (x <= 1.5e+136) {
		tmp = (((y_m * 2.0) + x) * (x - (y_m * 2.0))) / ((x * x) + (y_m * (y_m * 4.0)));
	} else {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (x <= 1.3e-126)
		tmp = Float64(fma(y_m, 2.0, x) / Float64(Float64(y_m * -2.0) + Float64(x * Float64(-1.0 - Float64(x / y_m)))));
	elseif (x <= 1.5e+136)
		tmp = Float64(Float64(Float64(Float64(y_m * 2.0) + x) * Float64(x - Float64(y_m * 2.0))) / Float64(Float64(x * x) + Float64(y_m * Float64(y_m * 4.0))));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[x, 1.3e-126], N[(N[(y$95$m * 2.0 + x), $MachinePrecision] / N[(N[(y$95$m * -2.0), $MachinePrecision] + N[(x * N[(-1.0 - N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.5e+136], N[(N[(N[(N[(y$95$m * 2.0), $MachinePrecision] + x), $MachinePrecision] * N[(x - N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * N[(y$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.3e-126

   1. Initial program 54.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 54.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 54.5%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 54.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 54.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 54.5%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 29.0%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 39.5%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 39.5%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 39.5%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 39.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 39.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 29.0%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 54.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 54.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 54.5%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 54.5%

         \[\leadsto \frac{\color{blue}{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. associate-/l* 55.6%

         \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{x + y \cdot 2}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      3. +-commutative 55.6%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{y \cdot 2 + x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. fma-define 55.6%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. *-commutative 55.6%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{y \cdot \left(y \cdot 4\right)}} \]

      6. associate-*r* 55.6%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot y\right) \cdot 4}} \]

      7. metadata-eval 55.6%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \left(y \cdot y\right) \cdot \color{blue}{\left(2 \cdot 2\right)}} \]

      8. swap-sqr 55.6%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      9. add-sqr-sqrt 55.6%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}}} \]

      10. hypot-undefine 55.6%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      11. hypot-undefine 55.6%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

      12. unpow2 55.6%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

   6. Applied egg-rr 55.6%

      \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 54.5%

         \[\leadsto \color{blue}{\frac{\left(x - y \cdot 2\right) \cdot \mathsf{fma}\left(y, 2, x\right)}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

      2. unpow2 54.5%

         \[\leadsto \frac{\left(x - y \cdot 2\right) \cdot \mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \mathsf{hypot}\left(x, y \cdot 2\right)}} \]

      3. frac-times 99.9%

         \[\leadsto \color{blue}{\frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

      4. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x - y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

      5. clear-num 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x - y \cdot 2}}} \]

      6. frac-times 100.0%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right) \cdot 1}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x - y \cdot 2}}} \]

      7. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(y, 2, x\right)}}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x - y \cdot 2}} \]

      8. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x - y \cdot 2}} \]

      9. sub-neg 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{\color{blue}{x + \left(-y \cdot 2\right)}}} \]

      10. distribute-rgt-neg-in 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + \color{blue}{y \cdot \left(-2\right)}}} \]

      11. metadata-eval 100.0%

          \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + y \cdot \color{blue}{-2}}} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \frac{\mathsf{hypot}\left(x, y \cdot 2\right)}{x + y \cdot -2}}} \]

   9. Taylor expanded in x around 0 58.3%

      \[\leadsto \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{-2 \cdot y + x \cdot \left(-1 \cdot \frac{x}{y} - 1\right)}} \]

   if 1.3e-126 < x < 1.49999999999999989e136

   1. Initial program 85.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 85.9%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 85.9%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 85.9%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 85.9%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 85.9%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 43.7%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 66.9%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 66.9%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 66.9%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 66.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 66.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 43.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 85.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 85.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 85.9%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 1.49999999999999989e136 < x

   1. Initial program 5.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 5.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 5.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 5.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 82.4%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 82.4%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. pow2 82.4%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 88.8%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 88.8%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 2, x\right)}{y \cdot -2 + x \cdot \left(-1 - \frac{x}{y}\right)}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := x - y\_m \cdot 2\\ \mathbf{if}\;x \cdot x \leq 1.5 \cdot 10^{-252}:\\ \;\;\;\;t\_0 \cdot \frac{0.5 + \frac{x}{y\_m} \cdot 0.25}{y\_m}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\left(y\_m \cdot 2 + x\right) \cdot t\_0}{x \cdot x + y\_m \cdot \left(y\_m \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (- x (* y_m 2.0))))
   (if (<= (* x x) 1.5e-252)
     (* t_0 (/ (+ 0.5 (* (/ x y_m) 0.25)) y_m))
     (if (<= (* x x) 5e+263)
       (/ (* (+ (* y_m 2.0) x) t_0) (+ (* x x) (* y_m (* y_m 4.0))))
       (+ 1.0 (* -8.0 (* (/ y_m x) (/ y_m x))))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = x - (y_m * 2.0);
	double tmp;
	if ((x * x) <= 1.5e-252) {
		tmp = t_0 * ((0.5 + ((x / y_m) * 0.25)) / y_m);
	} else if ((x * x) <= 5e+263) {
		tmp = (((y_m * 2.0) + x) * t_0) / ((x * x) + (y_m * (y_m * 4.0)));
	} else {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (y_m * 2.0d0)
    if ((x * x) <= 1.5d-252) then
        tmp = t_0 * ((0.5d0 + ((x / y_m) * 0.25d0)) / y_m)
    else if ((x * x) <= 5d+263) then
        tmp = (((y_m * 2.0d0) + x) * t_0) / ((x * x) + (y_m * (y_m * 4.0d0)))
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y_m / x) * (y_m / x)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = x - (y_m * 2.0);
	double tmp;
	if ((x * x) <= 1.5e-252) {
		tmp = t_0 * ((0.5 + ((x / y_m) * 0.25)) / y_m);
	} else if ((x * x) <= 5e+263) {
		tmp = (((y_m * 2.0) + x) * t_0) / ((x * x) + (y_m * (y_m * 4.0)));
	} else {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = x - (y_m * 2.0)
	tmp = 0
	if (x * x) <= 1.5e-252:
		tmp = t_0 * ((0.5 + ((x / y_m) * 0.25)) / y_m)
	elif (x * x) <= 5e+263:
		tmp = (((y_m * 2.0) + x) * t_0) / ((x * x) + (y_m * (y_m * 4.0)))
	else:
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(x - Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(x * x) <= 1.5e-252)
		tmp = Float64(t_0 * Float64(Float64(0.5 + Float64(Float64(x / y_m) * 0.25)) / y_m));
	elseif (Float64(x * x) <= 5e+263)
		tmp = Float64(Float64(Float64(Float64(y_m * 2.0) + x) * t_0) / Float64(Float64(x * x) + Float64(y_m * Float64(y_m * 4.0))));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = x - (y_m * 2.0);
	tmp = 0.0;
	if ((x * x) <= 1.5e-252)
		tmp = t_0 * ((0.5 + ((x / y_m) * 0.25)) / y_m);
	elseif ((x * x) <= 5e+263)
		tmp = (((y_m * 2.0) + x) * t_0) / ((x * x) + (y_m * (y_m * 4.0)));
	else
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(x - N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1.5e-252], N[(t$95$0 * N[(N[(0.5 + N[(N[(x / y$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+263], N[(N[(N[(N[(y$95$m * 2.0), $MachinePrecision] + x), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * N[(y$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.49999999999999997e-252

   1. Initial program 60.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 60.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 60.5%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 60.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 60.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 60.5%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 32.2%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 34.7%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 34.7%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 34.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 34.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 34.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 32.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 60.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 60.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 60.5%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 60.5%

         \[\leadsto \frac{\color{blue}{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. associate-/l* 61.3%

         \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{x + y \cdot 2}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      3. +-commutative 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{y \cdot 2 + x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. fma-define 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. *-commutative 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{y \cdot \left(y \cdot 4\right)}} \]

      6. associate-*r* 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot y\right) \cdot 4}} \]

      7. metadata-eval 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \left(y \cdot y\right) \cdot \color{blue}{\left(2 \cdot 2\right)}} \]

      8. swap-sqr 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      9. add-sqr-sqrt 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}}} \]

      10. hypot-undefine 61.3%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      11. hypot-undefine 61.3%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

      12. unpow2 61.3%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

   6. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

   7. Taylor expanded in y around inf 88.5%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \color{blue}{\frac{0.5 + 0.25 \cdot \frac{x}{y}}{y}} \]

   if 1.49999999999999997e-252 < (*.f64 x x) < 5.00000000000000022e263

   1. Initial program 83.1%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 83.1%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 83.2%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 83.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 83.2%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 83.2%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 44.1%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 67.9%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 67.9%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 67.9%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 67.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 67.9%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 44.1%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 83.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 83.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 83.2%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 5.00000000000000022e263 < (*.f64 x x)

   1. Initial program 5.6%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 5.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 5.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 82.0%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 82.0%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. pow2 82.0%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 88.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 88.2%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.5 \cdot 10^{-252}:\\ \;\;\;\;\left(x - y \cdot 2\right) \cdot \frac{0.5 + \frac{x}{y} \cdot 0.25}{y}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := y\_m \cdot \left(y\_m \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 1.5 \cdot 10^{-252}:\\ \;\;\;\;\left(x - y\_m \cdot 2\right) \cdot \frac{0.5 + \frac{x}{y\_m} \cdot 0.25}{y\_m}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* y_m (* y_m 4.0))))
   (if (<= (* x x) 1.5e-252)
     (* (- x (* y_m 2.0)) (/ (+ 0.5 (* (/ x y_m) 0.25)) y_m))
     (if (<= (* x x) 5e+263)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -8.0 (* (/ y_m x) (/ y_m x))))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = y_m * (y_m * 4.0);
	double tmp;
	if ((x * x) <= 1.5e-252) {
		tmp = (x - (y_m * 2.0)) * ((0.5 + ((x / y_m) * 0.25)) / y_m);
	} else if ((x * x) <= 5e+263) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (y_m * 4.0d0)
    if ((x * x) <= 1.5d-252) then
        tmp = (x - (y_m * 2.0d0)) * ((0.5d0 + ((x / y_m) * 0.25d0)) / y_m)
    else if ((x * x) <= 5d+263) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y_m / x) * (y_m / x)))
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = y_m * (y_m * 4.0);
	double tmp;
	if ((x * x) <= 1.5e-252) {
		tmp = (x - (y_m * 2.0)) * ((0.5 + ((x / y_m) * 0.25)) / y_m);
	} else if ((x * x) <= 5e+263) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = y_m * (y_m * 4.0)
	tmp = 0
	if (x * x) <= 1.5e-252:
		tmp = (x - (y_m * 2.0)) * ((0.5 + ((x / y_m) * 0.25)) / y_m)
	elif (x * x) <= 5e+263:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(y_m * Float64(y_m * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1.5e-252)
		tmp = Float64(Float64(x - Float64(y_m * 2.0)) * Float64(Float64(0.5 + Float64(Float64(x / y_m) * 0.25)) / y_m));
	elseif (Float64(x * x) <= 5e+263)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = y_m * (y_m * 4.0);
	tmp = 0.0;
	if ((x * x) <= 1.5e-252)
		tmp = (x - (y_m * 2.0)) * ((0.5 + ((x / y_m) * 0.25)) / y_m);
	elseif ((x * x) <= 5e+263)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(y$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1.5e-252], N[(N[(x - N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 + N[(N[(x / y$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+263], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.49999999999999997e-252

   1. Initial program 60.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 60.5%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 60.5%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 60.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 60.5%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 60.5%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 32.2%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 34.7%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 34.7%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 34.7%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 34.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 34.7%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 32.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 60.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 60.5%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 60.5%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 60.5%

         \[\leadsto \frac{\color{blue}{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. associate-/l* 61.3%

         \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{x + y \cdot 2}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      3. +-commutative 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{y \cdot 2 + x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. fma-define 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. *-commutative 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{y \cdot \left(y \cdot 4\right)}} \]

      6. associate-*r* 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot y\right) \cdot 4}} \]

      7. metadata-eval 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \left(y \cdot y\right) \cdot \color{blue}{\left(2 \cdot 2\right)}} \]

      8. swap-sqr 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      9. add-sqr-sqrt 61.3%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}}} \]

      10. hypot-undefine 61.3%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      11. hypot-undefine 61.3%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

      12. unpow2 61.3%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

   6. Applied egg-rr 61.3%

      \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

   7. Taylor expanded in y around inf 88.5%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \color{blue}{\frac{0.5 + 0.25 \cdot \frac{x}{y}}{y}} \]

   if 1.49999999999999997e-252 < (*.f64 x x) < 5.00000000000000022e263

   1. Initial program 83.1%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing

   if 5.00000000000000022e263 < (*.f64 x x)

   1. Initial program 5.6%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 5.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 5.6%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 5.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 82.0%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 82.0%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. pow2 82.0%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 88.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 88.2%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.5 \cdot 10^{-252}:\\ \;\;\;\;\left(x - y \cdot 2\right) \cdot \frac{0.5 + \frac{x}{y} \cdot 0.25}{y}\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 43000000:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m \cdot 2\right) \cdot \frac{0.5 + \frac{x}{y\_m} \cdot 0.25}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 43000000.0)
   (+ 1.0 (* -8.0 (* (/ y_m x) (/ y_m x))))
   (* (- x (* y_m 2.0)) (/ (+ 0.5 (* (/ x y_m) 0.25)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 43000000.0) {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	} else {
		tmp = (x - (y_m * 2.0)) * ((0.5 + ((x / y_m) * 0.25)) / y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 43000000.0d0) then
        tmp = 1.0d0 + ((-8.0d0) * ((y_m / x) * (y_m / x)))
    else
        tmp = (x - (y_m * 2.0d0)) * ((0.5d0 + ((x / y_m) * 0.25d0)) / y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 43000000.0) {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	} else {
		tmp = (x - (y_m * 2.0)) * ((0.5 + ((x / y_m) * 0.25)) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 43000000.0:
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)))
	else:
		tmp = (x - (y_m * 2.0)) * ((0.5 + ((x / y_m) * 0.25)) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 43000000.0)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	else
		tmp = Float64(Float64(x - Float64(y_m * 2.0)) * Float64(Float64(0.5 + Float64(Float64(x / y_m) * 0.25)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 43000000.0)
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	else
		tmp = (x - (y_m * 2.0)) * ((0.5 + ((x / y_m) * 0.25)) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 43000000.0], N[(1.0 + N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 + N[(N[(x / y$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3e7

   1. Initial program 57.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 57.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 58.6%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 58.6%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. pow2 58.6%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 62.1%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 62.1%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   if 4.3e7 < y

   1. Initial program 47.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 47.4%

         \[\leadsto \frac{x \cdot x - \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot y} \cdot \sqrt{\left(y \cdot 4\right) \cdot y}}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. difference-of-squares 47.4%

         \[\leadsto \frac{\color{blue}{\left(x + \sqrt{\left(y \cdot 4\right) \cdot y}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. *-commutative 47.4%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. associate-*r* 47.4%

         \[\leadsto \frac{\left(x + \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. sqrt-prod 47.4%

         \[\leadsto \frac{\left(x + \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. sqrt-unprod 47.2%

         \[\leadsto \frac{\left(x + \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      7. add-sqr-sqrt 47.4%

         \[\leadsto \frac{\left(x + \color{blue}{y} \cdot \sqrt{4}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      8. metadata-eval 47.4%

         \[\leadsto \frac{\left(x + y \cdot \color{blue}{2}\right) \cdot \left(x - \sqrt{\left(y \cdot 4\right) \cdot y}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      9. *-commutative 47.4%

         \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      10. associate-*r* 47.4%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      11. sqrt-prod 47.4%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{4}}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      12. sqrt-unprod 47.2%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      13. add-sqr-sqrt 47.4%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - \color{blue}{y} \cdot \sqrt{4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      14. metadata-eval 47.4%

          \[\leadsto \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot \color{blue}{2}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. Applied egg-rr 47.4%

      \[\leadsto \frac{\color{blue}{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   5. Step-by-step derivation

      1. *-commutative 47.4%

         \[\leadsto \frac{\color{blue}{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. associate-/l* 48.9%

         \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{x + y \cdot 2}{x \cdot x + \left(y \cdot 4\right) \cdot y}} \]

      3. +-commutative 48.9%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{y \cdot 2 + x}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. fma-define 48.9%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(y, 2, x\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. *-commutative 48.9%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{y \cdot \left(y \cdot 4\right)}} \]

      6. associate-*r* 48.9%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot y\right) \cdot 4}} \]

      7. metadata-eval 48.9%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \left(y \cdot y\right) \cdot \color{blue}{\left(2 \cdot 2\right)}} \]

      8. swap-sqr 48.9%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{x \cdot x + \color{blue}{\left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      9. add-sqr-sqrt 48.8%

         \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}}} \]

      10. hypot-undefine 48.9%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \sqrt{x \cdot x + \left(y \cdot 2\right) \cdot \left(y \cdot 2\right)}} \]

      11. hypot-undefine 48.9%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right) \cdot \color{blue}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]

      12. unpow2 48.9%

          \[\leadsto \left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\color{blue}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

   6. Applied egg-rr 48.9%

      \[\leadsto \color{blue}{\left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{{\left(\mathsf{hypot}\left(x, y \cdot 2\right)\right)}^{2}}} \]

   7. Taylor expanded in y around inf 83.7%

      \[\leadsto \left(x - y \cdot 2\right) \cdot \color{blue}{\frac{0.5 + 0.25 \cdot \frac{x}{y}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 43000000:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot 2\right) \cdot \frac{0.5 + \frac{x}{y} \cdot 0.25}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 44000000:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y\_m}{x} \cdot \frac{y\_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 44000000.0) (+ 1.0 (* -8.0 (* (/ y_m x) (/ y_m x)))) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 44000000.0) {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 44000000.0d0) then
        tmp = 1.0d0 + ((-8.0d0) * ((y_m / x) * (y_m / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 44000000.0) {
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 44000000.0:
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 44000000.0)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y_m / x) * Float64(y_m / x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 44000000.0)
		tmp = 1.0 + (-8.0 * ((y_m / x) * (y_m / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 44000000.0], N[(1.0 + N[(-8.0 * N[(N[(y$95$m / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.4e7

   1. Initial program 57.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 57.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 58.6%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 58.6%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. pow2 58.6%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 62.1%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   7. Applied egg-rr 62.1%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   if 4.4e7 < y

   1. Initial program 47.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 47.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 83.3%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 50000000.0) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 50000000.0d0) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 50000000.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 50000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 50000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 50000000.0], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 5e7

   1. Initial program 57.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 57.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 57.3%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 60.6%

      \[\leadsto \color{blue}{1} \]

   if 5e7 < y

   1. Initial program 47.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 47.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. distribute-rgt-neg-in 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. fma-define 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

      7. *-commutative 47.4%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

   3. Simplified 47.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 83.3%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 50.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

y_m = fabs(y);
double code(double x, double y_m) {
	return -1.0;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = -1.0d0
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return -1.0;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

y_m = abs(y)
function code(x, y_m)
	return -1.0
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = -1.0;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 55.1%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
2. Step-by-step derivation

   1. fma-neg 55.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. *-commutative 55.1%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   3. distribute-rgt-neg-in 55.1%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(-y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   4. distribute-rgt-neg-in 55.1%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \color{blue}{\left(y \cdot \left(-4\right)\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   5. metadata-eval 55.1%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot \color{blue}{-4}\right)\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   6. fma-define 55.1%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}} \]

   7. *-commutative 55.1%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)} \]

3. Simplified 55.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 50.1%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Developer Target 1: 51.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t\_0\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t\_3}{x \cdot x + t\_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t\_1} - t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t\_1}}\right)}^{2} - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024148 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 9743233849626781/10000000000000000) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4))))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))